The wind integrated multi-energy systems (MES) have gained significant momentum in recent years on account of their self-sufficiency and attractive clean attributes. This study puts forward a bi-level multi-follower optimization framework to study the tactical response of a wind integrated MES in the wholesale electricity market (WEM) and the natural gas market (NGM) as a price setter. At the upper level, the MES endeavors to minimize the overall operational costs by giving the best offer/bid in WEM/NGM, and by utilizing thermal energy storage (TES), compressed air energy storage (CAES), and natural gas storage (NGS). When the MES submits offers/bids in WEM and NGM, the NGM and WEM operators, as individual followers, clear their respective markets to maximize public welfare and announce the ultimate market-clearing price (MCP). Additionally, risk-averse and risk-seeker information gap decision theory (IGDT) have been deployed to provide various decision-making options for MES operators considering wind underproduction and overproduction scenarios. Standard 6-node natural gas network (NGN) and 6-bus transmission system (TS) have been deployed to model WEM and NGM, respectively. The results testify to the capabilities of the MES in influencing the decisions of WEM and NGM.